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Advance Physics Letter
________________________________________________________________________________
ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014
43
Determination of Optical Constants of SnO2 Thin Film Using Matlab
Curve Fitting Method
Chitrakant Sharma1
, Gajendra Singh Rathore2
, Vikas Dubey3
1.2
Department of Electronics and Communication Engineering, School of Engineering & IT, MATS University, Gullu,
Arang (C.G.), India
3
Department of Physics, School of Engineering & IT, MATS University, Gullu, Arang (C.G.), India
Email : sharma.chitrakant@gmail.com, gajendra05in@gmail.com
Abstract:- Thin absorbing films are becoming common in
optical coating applications. Such films require proper
techniques to insure unique results for both thickness and
optical constants. This project describes the determination of
optical constant and thickness of SnO2 thin film using
MATLAB curve fitting method. Various techniques are
available for determining the properties of surfaces and thin
films. A versatile method for determination of the optical
constants is described that can be applied to a variety of
coating materials. It is based on the use of an optical thin
film synthesis program to adjust the constants of dispersion
equations until a good fit is obtained between measured and
calculated spectral transmittance and/or reflectance curves.
A method for fitting the optical transmission data using
MATLAB software is presented in this paper. The sensitivity
of the determination can be increased by a suitable
combination of measurement quantities. Because more than
the minimum amount of data can be used, sensitivity to
measurement errors and the chances of obtaining multiple
solutions can both be reduced. This is a multiple parameter,
nonlinear-curve fitting method which is used to determine
the optical constants and thickness of SnO2 thin films. The
transmission equation is customized and six constants are
determined using the MATLAB curve-fitting tool box.
Keywords: Thin Film, Spectral Transmittance or
Reflectance, Optical Constants, Complex Refractive Index,
Dielectric Function.
INTRODUCTION
In current years there has been an intensifying need for
an accurate data of the optical constants n and k of thin
absorbing films over a wavelength range. Since the
optical constants of the materials have generally been
completely unknown or determined only in a narrow
wavelength range, proposed designs for such films have
had to be tested experimentally rather than by the much
more rapid method of numerical simulation. The
insufficient knowledge of optical constants has thus
slowed progress in its application. The refractive index
of a material is the most important property of any
optical system that uses refraction. It is used to calculate
the focusing power of lenses, and the dispersive power
of prisms. Since refractive index is a fundamental
physical property of a substance, it is often used to
identify a particular substance, confirm its purity, or
measure its concentration. Refractive index is used to
measure solids (glasses and gemstones), liquids, and
gases. Most commonly it is used to measure the
concentration of a solute in an aqueoussolution. A
refractometer is the instrument used to measure
refractive index. For a solution of sugar, the refractive
index can be used to determine the sugar content. In
GPS, the index of refraction is utilized in ray-tracing to
account for the radio propagation delay due to the
Earth's electrically neutral atmosphere. It is also used in
Satellite link design for the Computation of radiowave
attenuation in the atmosphere.
OPTICAL CONSTANTS AND THICKNESS
OF THIN FILM USING MATLAB
The optical constants n(λ) and α(λ) are of particular
significance in knowing the optical properties of thin
films. In this method measurement of the transmittance
T of light through a parallel-faced dielectric film is
sufficient to determine the real and imaginary parts of
the complex refractive index and the thickness of thin
film. Optical constants are usually determined using the
envelope method where extreme values of transmission
data are used. A multiparameter curve-fitting method
was proposed by Xuantong (1990) for determining
optical constants (refractive index n and absorption
coefficient α) and the thickness„t‟ of optical films from
transmission spectra. The curve-fitting method using
MATLAB software is used for determining the optical
constants. MATLAB provides a platform to perform
parametric data fitting using toolbox- library equations
or using a custom equation. Custom equation can be
defined to suit the specific curve fitting requirements.
The fit statistics are used to determine the goodness of
fit. Analysis capabilities such as extrapolation,
Advance Physics Letter
________________________________________________________________________________
ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014
44
differentiation and integration are other helpful features
in using this toolbox. The transmission equation for
uniform thickness and homogeneous film is made use of
in the calculations. A non-linear least-squares-fitting has
been used to fit the experimental transmittance data on
the basis of dispersion models for refractive index and
absorption. The goodness-of-fit results are helpful in the
fitting routine to improve the accuracy of the result.
Once the equation of transmission is customized, the
same equation can be used for all samples by changing
the boundary conditions for making the fitting procedure
fast and reliable.
Curve fitting
In this homogeneous model, the transmittance T of
weakly absorbing films can be expressed by the
following equation
)/4cos(2
16
21
22
2
2
1
2
ntACCACC
Annn
T so


---- (1)
Where
)(
)(
1
nn
nn
C
s
o



)(
)(
2
nn
nn
C
s
o



A = exp (-4πkt/λ) = exp (-αt)
And ns is the refractive index of substrate, no is the
refractive index of ambient (air), and λ is the wavelength
of the optical beam in air. The dispersion equation for n
is the Cauchy‟s formula
n = a + b/ λ2 ---- (2)
the form chosen for α is an exponential law,
α = d exp(-eλ) +f ---- (3)
and the thickness „t‟ is taken to be the fit parameter „c‟.
The transmission curve of each specimen was fitted with
equation 1 and the six parameters „a to f‟ were
determined using curve fitting toolbox. The custom
equation given in the toolbox is as follows:
fittedmodel1(x)=(24.*(a+(b./x.^2)).^2.*exp(-(d.*exp(-
e.*x).*c)-
(f.*c)))./((((a+(b./x.^2))+1).^2.*((a+(b./x.^2))+1.5).^2)+
((((a+(b./x.^2))-1).^2.*(1.5-(a+(b./x.^2))).^2).*(exp(-
(d.*exp(-e.*x).*c)-(f.*c))).^2)-(2.*((a+(b./x.^2)).^2-
1).*((a+(b./x.^2)).^2-2.25).*exp(-(d.*exp(-e.*x).*c)-
(f.*c)).*cos((4.*pi.*(a+(b./x.^2)).*c)./x)))
To determine the best fit, both the graphical and
numerical fit results are examined. In the graphical
approach the transmission curve and the fitted curve are
compared and the two should overlap each other for
most of the points. The residuals are displayed
graphically as a line plot and the residual from a good fit
should look random with no apparent pattern. A pattern,
such as a tendency for consecutive residuals to have the
same sign, can be an indication that better model exists.
As the graphical results cannot examine fully the
goodness of a fit, the numerical fit results should be
considered to evaluate the fit results. Two types of
numerical fit results are displayed in the results. The
first is the goodness of fit statistics which determines
how well the curve fits the data. The confidence
intervals on the coefficients determine their accuracy.
The sum of squares due to error (SSE) and adjusted R-
square statistics are useful in determining the best fit.
The SSE value closer to zero indicates a better fit and
the adjusted R-square value should be closer to 1 for the
best fit. The parameters are determined to 95%
confidence bound and the prediction bounds give the
accuracy of the parameters. If there is a wider
prediction bounds for the parameters then the parameters
are not very accurate.
RESULT AND DISCUSSION
The fitted and transmission curve of the samples used in
the present work are shown below.
Figure1: The fitted transmission spectrum of sample
SnO2
Table1: The six parameters determined with custom
equation using curve fitting toolbox
Sample a b (x10
-14
)
c
(x10
-7)
d (
x10 5
)
e ( x10
6
)
f (
x10 3
)
R2
SnO2 1.518 0.330 2.52 0.039 0.5586 2.738 0.9875
The constant „a‟ is the value of the refractive index in
the longer wavelength region and the constant „c‟ is the
thickness of the sample. The sum of squares due to error
(SSE) and adjusted R-square statistics are useful in
determining the best fit. The SSE value closer to zero
indicates a better fit and the adjusted R-square value
should be closer to 1 for the best fit.
Result of absorption coefficient (α) and extinction
coefficient (k) values which obtain from Beer’s law
Advance Physics Letter
________________________________________________________________________________
ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014
45
S.No.
Wavelength
(nm)
Absorption
coefficient(α)
Extinction
coefficient(k)
1. 532 160 CM-1
6.8096 × 10-4
2. 532 155 CM-1
6.5968 × 10-4
3. 532 167 CM-1
7.10752 ×10-4
Reflectance and transmittance spectra of film using a
MATLAB program
For a single thin film on a substrate we get the amplitude
reflectance, r, and the amplitude transmittance, t. At
normal incidence the following applies:




 i
i
err
err
r 2
21
2
21
1
; 21
21
1
NN
NN
r



,
32
32
2
NN
NN
r



t 
t1t2e i
1 r1r2e2 i
; 21
1
1
2
NN
N
t


,
t2 
2N2
N2  N3
Where the phase shift, ∂, is given by
  d
N
dN



2
22
2
incidencenormalcos
2 



And r1 and t1 are the Fresnel coefficients for the
boundary between air and film and r2 and t2 are the
corresponding coefficients for the boundary between
film and substrate.
With the equations above we can calculate the intensity
reflectance, R, and the intensity transmittance, T,
according to
*rrR  and
*Re
1
3
tt
N
N
T 






In our case, the medium through which the light is
incident is air, i.e. N1=1
Figure2: Transmission and Reflectance spectra of SnO2
thin film
CONCLUSION
In this project optical constant and thickness of SnO2
thin film has been measured using MATLAB curve
fitting method. It is quite simple to calculate R and T for
a single film on a non-absorbing substrate using this
method and to study how variations in the optical
constants n and k of thin films and substrates influence
R and T spectra? In order to study interference effects
only, we look at dispersion free materials for which n
and k are not wavelength dependent.
REFERENCES
[1] Manifacier, J.C., Gasiot, J., Fillard, J.P., (1976)
“A simple method for the determination of the
optical cnstants n, k and the thickness of a
weakly absorbing thin film” J. of Phy E. Sci.
Instrum, 9, pp 1002-1007.
[2] Xuantong Ying, Albert Feldman, Farabaugh,
E.N., (1990) “Fitting of transmission data for
determining the optical constants and thicknesses
of optical films” J.Appl. Phy.67(4), pp 2056-
2063
[3] Swanepoel, R., (1983) “Determination of the
thickness and optical constants of amorphous
silicon” J. Phys. E. Sci. Instrum., Vol.16, pp
1214-1221
[4] Born, Wolf, (1985) “Principles of Optics”
Pergamon Press.
[5] Mártil, I., Díaz, G.G., (1992) “Undergraduate
laboratory experiment: measurement of the
complex refractive index and the band gap of a
thin film semiconductor” Am. J. Phys., 60 (1), pp
83-89.
[6] R. M. A. Azzam and N. M. Bashara,
Ellipsometry and Polarized Light (North-
Holland, Amsterdam, 1977), pp. 255-262.
[7] D. E. Aspnes and A. A. Studna, Phys. Rev. B 27,
985 (1983).
200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lambda (nm)
d=250, N1=1, N2=1.517 och N3=1.46
RochT
R
T
Advance Physics Letter
________________________________________________________________________________
ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014
46
[8] W. R. Hunter, Appl. Opt. 21, 2103 (1982).
[9] H. M. Liddell, Computer Aided Techniques for
the Design of Multilayer Filters pp.
[10] A. Hjortsberg, Appl. Opt. 20, 1254 (1981).
[11] H.G. Tompkins, E.A. Irene (Eds.), Handbook of
Ellipsometry, William Andrew Publishing,
Norwich NY 2005.
[12] H. Fujiwara, Spectroscopic Ellipsometry
Principles and Applications, John Wiley & Sons
Ltd, West Sussex, England, 2007.
[13] M. H. Habibi, N. Talebian Acta Chim. Solv. 52
(2005) 53.
[14] Brian G. Lewis and David C. Paine, MRS
Bulletin. Aug (2000) 22.
[15] M. Yamaguchi, Ari Ide Eldessabi, H. Nomura
and N. Yasui, Thin Solid Films, 447 (2004) 1 15.


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  • 1. Advance Physics Letter ________________________________________________________________________________ ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014 43 Determination of Optical Constants of SnO2 Thin Film Using Matlab Curve Fitting Method Chitrakant Sharma1 , Gajendra Singh Rathore2 , Vikas Dubey3 1.2 Department of Electronics and Communication Engineering, School of Engineering & IT, MATS University, Gullu, Arang (C.G.), India 3 Department of Physics, School of Engineering & IT, MATS University, Gullu, Arang (C.G.), India Email : sharma.chitrakant@gmail.com, gajendra05in@gmail.com Abstract:- Thin absorbing films are becoming common in optical coating applications. Such films require proper techniques to insure unique results for both thickness and optical constants. This project describes the determination of optical constant and thickness of SnO2 thin film using MATLAB curve fitting method. Various techniques are available for determining the properties of surfaces and thin films. A versatile method for determination of the optical constants is described that can be applied to a variety of coating materials. It is based on the use of an optical thin film synthesis program to adjust the constants of dispersion equations until a good fit is obtained between measured and calculated spectral transmittance and/or reflectance curves. A method for fitting the optical transmission data using MATLAB software is presented in this paper. The sensitivity of the determination can be increased by a suitable combination of measurement quantities. Because more than the minimum amount of data can be used, sensitivity to measurement errors and the chances of obtaining multiple solutions can both be reduced. This is a multiple parameter, nonlinear-curve fitting method which is used to determine the optical constants and thickness of SnO2 thin films. The transmission equation is customized and six constants are determined using the MATLAB curve-fitting tool box. Keywords: Thin Film, Spectral Transmittance or Reflectance, Optical Constants, Complex Refractive Index, Dielectric Function. INTRODUCTION In current years there has been an intensifying need for an accurate data of the optical constants n and k of thin absorbing films over a wavelength range. Since the optical constants of the materials have generally been completely unknown or determined only in a narrow wavelength range, proposed designs for such films have had to be tested experimentally rather than by the much more rapid method of numerical simulation. The insufficient knowledge of optical constants has thus slowed progress in its application. The refractive index of a material is the most important property of any optical system that uses refraction. It is used to calculate the focusing power of lenses, and the dispersive power of prisms. Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids (glasses and gemstones), liquids, and gases. Most commonly it is used to measure the concentration of a solute in an aqueoussolution. A refractometer is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content. In GPS, the index of refraction is utilized in ray-tracing to account for the radio propagation delay due to the Earth's electrically neutral atmosphere. It is also used in Satellite link design for the Computation of radiowave attenuation in the atmosphere. OPTICAL CONSTANTS AND THICKNESS OF THIN FILM USING MATLAB The optical constants n(λ) and α(λ) are of particular significance in knowing the optical properties of thin films. In this method measurement of the transmittance T of light through a parallel-faced dielectric film is sufficient to determine the real and imaginary parts of the complex refractive index and the thickness of thin film. Optical constants are usually determined using the envelope method where extreme values of transmission data are used. A multiparameter curve-fitting method was proposed by Xuantong (1990) for determining optical constants (refractive index n and absorption coefficient α) and the thickness„t‟ of optical films from transmission spectra. The curve-fitting method using MATLAB software is used for determining the optical constants. MATLAB provides a platform to perform parametric data fitting using toolbox- library equations or using a custom equation. Custom equation can be defined to suit the specific curve fitting requirements. The fit statistics are used to determine the goodness of fit. Analysis capabilities such as extrapolation,
  • 2. Advance Physics Letter ________________________________________________________________________________ ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014 44 differentiation and integration are other helpful features in using this toolbox. The transmission equation for uniform thickness and homogeneous film is made use of in the calculations. A non-linear least-squares-fitting has been used to fit the experimental transmittance data on the basis of dispersion models for refractive index and absorption. The goodness-of-fit results are helpful in the fitting routine to improve the accuracy of the result. Once the equation of transmission is customized, the same equation can be used for all samples by changing the boundary conditions for making the fitting procedure fast and reliable. Curve fitting In this homogeneous model, the transmittance T of weakly absorbing films can be expressed by the following equation )/4cos(2 16 21 22 2 2 1 2 ntACCACC Annn T so   ---- (1) Where )( )( 1 nn nn C s o    )( )( 2 nn nn C s o    A = exp (-4πkt/λ) = exp (-αt) And ns is the refractive index of substrate, no is the refractive index of ambient (air), and λ is the wavelength of the optical beam in air. The dispersion equation for n is the Cauchy‟s formula n = a + b/ λ2 ---- (2) the form chosen for α is an exponential law, α = d exp(-eλ) +f ---- (3) and the thickness „t‟ is taken to be the fit parameter „c‟. The transmission curve of each specimen was fitted with equation 1 and the six parameters „a to f‟ were determined using curve fitting toolbox. The custom equation given in the toolbox is as follows: fittedmodel1(x)=(24.*(a+(b./x.^2)).^2.*exp(-(d.*exp(- e.*x).*c)- (f.*c)))./((((a+(b./x.^2))+1).^2.*((a+(b./x.^2))+1.5).^2)+ ((((a+(b./x.^2))-1).^2.*(1.5-(a+(b./x.^2))).^2).*(exp(- (d.*exp(-e.*x).*c)-(f.*c))).^2)-(2.*((a+(b./x.^2)).^2- 1).*((a+(b./x.^2)).^2-2.25).*exp(-(d.*exp(-e.*x).*c)- (f.*c)).*cos((4.*pi.*(a+(b./x.^2)).*c)./x))) To determine the best fit, both the graphical and numerical fit results are examined. In the graphical approach the transmission curve and the fitted curve are compared and the two should overlap each other for most of the points. The residuals are displayed graphically as a line plot and the residual from a good fit should look random with no apparent pattern. A pattern, such as a tendency for consecutive residuals to have the same sign, can be an indication that better model exists. As the graphical results cannot examine fully the goodness of a fit, the numerical fit results should be considered to evaluate the fit results. Two types of numerical fit results are displayed in the results. The first is the goodness of fit statistics which determines how well the curve fits the data. The confidence intervals on the coefficients determine their accuracy. The sum of squares due to error (SSE) and adjusted R- square statistics are useful in determining the best fit. The SSE value closer to zero indicates a better fit and the adjusted R-square value should be closer to 1 for the best fit. The parameters are determined to 95% confidence bound and the prediction bounds give the accuracy of the parameters. If there is a wider prediction bounds for the parameters then the parameters are not very accurate. RESULT AND DISCUSSION The fitted and transmission curve of the samples used in the present work are shown below. Figure1: The fitted transmission spectrum of sample SnO2 Table1: The six parameters determined with custom equation using curve fitting toolbox Sample a b (x10 -14 ) c (x10 -7) d ( x10 5 ) e ( x10 6 ) f ( x10 3 ) R2 SnO2 1.518 0.330 2.52 0.039 0.5586 2.738 0.9875 The constant „a‟ is the value of the refractive index in the longer wavelength region and the constant „c‟ is the thickness of the sample. The sum of squares due to error (SSE) and adjusted R-square statistics are useful in determining the best fit. The SSE value closer to zero indicates a better fit and the adjusted R-square value should be closer to 1 for the best fit. Result of absorption coefficient (α) and extinction coefficient (k) values which obtain from Beer’s law
  • 3. Advance Physics Letter ________________________________________________________________________________ ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014 45 S.No. Wavelength (nm) Absorption coefficient(α) Extinction coefficient(k) 1. 532 160 CM-1 6.8096 × 10-4 2. 532 155 CM-1 6.5968 × 10-4 3. 532 167 CM-1 7.10752 ×10-4 Reflectance and transmittance spectra of film using a MATLAB program For a single thin film on a substrate we get the amplitude reflectance, r, and the amplitude transmittance, t. At normal incidence the following applies:      i i err err r 2 21 2 21 1 ; 21 21 1 NN NN r    , 32 32 2 NN NN r    t  t1t2e i 1 r1r2e2 i ; 21 1 1 2 NN N t   , t2  2N2 N2  N3 Where the phase shift, ∂, is given by   d N dN    2 22 2 incidencenormalcos 2     And r1 and t1 are the Fresnel coefficients for the boundary between air and film and r2 and t2 are the corresponding coefficients for the boundary between film and substrate. With the equations above we can calculate the intensity reflectance, R, and the intensity transmittance, T, according to *rrR  and *Re 1 3 tt N N T        In our case, the medium through which the light is incident is air, i.e. N1=1 Figure2: Transmission and Reflectance spectra of SnO2 thin film CONCLUSION In this project optical constant and thickness of SnO2 thin film has been measured using MATLAB curve fitting method. It is quite simple to calculate R and T for a single film on a non-absorbing substrate using this method and to study how variations in the optical constants n and k of thin films and substrates influence R and T spectra? In order to study interference effects only, we look at dispersion free materials for which n and k are not wavelength dependent. REFERENCES [1] Manifacier, J.C., Gasiot, J., Fillard, J.P., (1976) “A simple method for the determination of the optical cnstants n, k and the thickness of a weakly absorbing thin film” J. of Phy E. Sci. Instrum, 9, pp 1002-1007. [2] Xuantong Ying, Albert Feldman, Farabaugh, E.N., (1990) “Fitting of transmission data for determining the optical constants and thicknesses of optical films” J.Appl. Phy.67(4), pp 2056- 2063 [3] Swanepoel, R., (1983) “Determination of the thickness and optical constants of amorphous silicon” J. Phys. E. Sci. Instrum., Vol.16, pp 1214-1221 [4] Born, Wolf, (1985) “Principles of Optics” Pergamon Press. [5] Mártil, I., Díaz, G.G., (1992) “Undergraduate laboratory experiment: measurement of the complex refractive index and the band gap of a thin film semiconductor” Am. J. Phys., 60 (1), pp 83-89. [6] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North- Holland, Amsterdam, 1977), pp. 255-262. [7] D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983). 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Lambda (nm) d=250, N1=1, N2=1.517 och N3=1.46 RochT R T
  • 4. Advance Physics Letter ________________________________________________________________________________ ISSN (Print) : 2349-1094, ISSN (Online) : 2349-1108, Vol_1, Issue_1, 2014 46 [8] W. R. Hunter, Appl. Opt. 21, 2103 (1982). [9] H. M. Liddell, Computer Aided Techniques for the Design of Multilayer Filters pp. [10] A. Hjortsberg, Appl. Opt. 20, 1254 (1981). [11] H.G. Tompkins, E.A. Irene (Eds.), Handbook of Ellipsometry, William Andrew Publishing, Norwich NY 2005. [12] H. Fujiwara, Spectroscopic Ellipsometry Principles and Applications, John Wiley & Sons Ltd, West Sussex, England, 2007. [13] M. H. Habibi, N. Talebian Acta Chim. Solv. 52 (2005) 53. [14] Brian G. Lewis and David C. Paine, MRS Bulletin. Aug (2000) 22. [15] M. Yamaguchi, Ari Ide Eldessabi, H. Nomura and N. Yasui, Thin Solid Films, 447 (2004) 1 15. 